Mourinho controversially signed Cole at the end of his contract at Arsenal in 2006. Cole, reunited with Mourinho following the Portuguese's return to Stamford Bridge, thinks any player would find it difficult to turn down the Chelsea boss.
"I jumped at the chance," England defender Cole said. "He's a great manager. He's proved how good he is with the number of trophies he's won with different teams.
"He's great at motivating players and keeping you on your toes, keeping you doing what you should be doing.
"For me, if there's a player he wants to come to Chelsea, it would be one thought in my mind: to come here and to be under a great manager. It's a pleasure to play under him."
The comments are sure to interest Wayne Rooney, who is Mourinho's sole transfer target.
Chelsea saw a bid for the Manchester United striker rejected last week.
Cole has won every honour in the club game since joining Chelsea. The 32-year-old hopes to give Mourinho strong service in the forthcoming campaign, believing he was below his best under the former Porto, Inter Milan and Real Madrid boss in the Portuguese's first spell in west London.
Cole added: "Having the manager back now from 2006 when he bought me, I still feel I owe him something. I didn't really feel I played as well as I could and as well as I have for Chelsea (since), when he bought me.
"On a personal level, I think I owe him a lot. As a team, I hope we can go on and win many trophies. It's a big club and we need to win trophies.
"With the new players we've got in, we're training hard and we're confident that, hopefully, this season could be a great season for Chelsea again.
"I've been lucky enough to win big trophies. Of course, I hope to win more. Having the manager back, hopefully we can do it all together. As players we owe it to the manager to win the league or the Champions League again."
Cole earlier this year signed a one-year contract extension which incorporates the forthcoming season.
Asked if he would like to stay on longer, he said: "I hope so.
"That's not down to me. I've just got to keep training hard, try to keep playing well and hopefully keep my place in the team.
"We'll see what goes from there. I love Chelsea. I've been here seven years now and I would love to stay here as long as I can."